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For example, consider the three components trivariate finite mixture model with following restricted covariance structure. Y Y Y 1 2 3 = X = X 1 = X 2 3 + X + X + X 12 12 13 + X + X + X 13 23 23 ⎡1 0 0 1 1 0 ⎤ A = ⎢ ⎥ ⎢ 0 1 0 1 0 1 ⎥ and ⎢⎣ 0 0 1 0 1 1 ⎥⎦ p , 1 p and p 2 3 are mixing proportions. The element wise expectation of matrix B (λ) are E B ) , E B ) and E B ) respectively. Then ( 1 ( 2 ( 3 E ( B) = p1E( B1) + p2E( B 2 ) + p3E( B3) 2 ⎡E( λ11) + E( λ11 ) E( λλ 11 21) ... E( λλ 11 1) ⎤ t ⎢ 2 ⎥ E( λλ 11 21) E( λ21) + E( λ21) ... E( λλ 21 t1) E( B 1) = ⎢ ⎥ ⎢ ... ⎥ ⎢ 2 ⎥ ⎢⎣ E( λ11λt1) ... E( λt1) + E( λt1) ⎥⎦ 2 ⎡E( λ12) + E( λ12) E( λ12λ22) ... E( λ12λ 2) ⎤ t ⎢ 2 ⎥ E( λ12λ22) E( λ22) + E( λ22) ... E( λ22λt 2) E( B 2) = ⎢ ⎥ ⎢ ... ⎥ ⎢ 2 ⎥ ⎢⎣ E( λ12λt2 ) ... E( λt2 ) + E( λt2 ) ⎥⎦ 2 ⎡E( λ13) + E( λ13) E( λ13λ23) ... E( λ13λ 3) ⎤ t ⎢ 2 ⎥ E( λ13λ23) E( λ23) + E( λ23) ... E( λ23λt 3) E( B 3) = ⎢ ⎥ ⎢ ... ⎥ ⎢ 2 ⎥ ⎢⎣ E( λ13λt3 ) ... E( λt3) + E( λt3) ⎥⎦ 144
and E ( Y ) = AM where ⎡λ1 ⎤ ⎢ ⎥ ⎢ λ2 M = ⎥ . ⎢ ... ⎥ ⎢ ⎥ ⎣λ t ⎦ Details of the proof of the unconditional variance of vector Y are given in Karlis and Meligkotsidou (2006). A brief description of the multivariate Poisson-log normal distribution is given in the next section, and these models were compared with the finite mixture models in section 7.5. 7.4 Multivariate Poisson-log Normal distribution The multivariate Poisson-log normal distribution (Aitchison and Ho, 1989) is a natural extension of the univariate Poisson-log normal distribution. Here the mixing of d independent Poisson distributions Po( λ i ) is achieved by placing a d-dimensional lognormal distribution on the d-dimensional vector λ . The multivariate Poisson-log normal distribution supports negative and positive correlation between the count variables. 7.4.1 Definition and the properties Let g( λ | µ , Σ) denote the probability density function of the d-dimensional log normal distribution Λ d ( µ , Σ) , so that 145
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MULTIVARIATE POISSON HIDDEN MARKOV
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ABSTRACT Multivariate count data ar
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ACKNOWLEDGEMENT First I would like
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TABLE OF CONTENTS PERMISSION TO USE
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6.4 Data analysis..................
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Table 7.7: Loglikelihood and AIC to
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Figure 6.10: Loglikelihood, AIC and
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CHAPTER 1 GENERAL INTRODUCTION 1.1
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North Carolina, is modelled by Symo
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concept of Markov models to include
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egularity constraints on the underl
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CHAPTER 2 HIDDEN MARKOV MODELS ( HM
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Given the coin tossing experiment,
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P 11 P 22 P 12 1 2 P 21 P 32 P 13 P
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0.8 0.6 0.2 1 0.4 2 (1) P [H]=2/3 (
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… Urn 1 Urn 2 Urn N P[Red]= b 1 (
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5. The probability distribution of
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focus of this section. Random field
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Now consider a random field { X ( s
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for some real β . Again, the denom
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Since original HMMs were designed a
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CHAPTER 3 INFERENCE IN HIDDEN MARKO
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sequence. If we have several compet
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Using this equation we can calculat
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3.2.2 Problem 2 and its solution Gi
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Letting Ut( S1 = i1, S2 = i2,..., S
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ξ α () iPβ ( jb ) ( y ) t ij t+
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ˆ ( n) = b j Expected Number of ti
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CHAPTER 4 HIDDEN MARKOV MODEL AND T
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4.2.1 Wild Oats Figure 4.1: Wild Oa
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4.2.2.1 Effects on crop quality Wil
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at each of the 150 grid locations (
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In the literature review (section 1
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estimated through the observations
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CHAPTER 5 MULTIVARIATE POISSON DIST
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Y Y Y 1 2 3 = X = X 1 = X 2 3 + X +
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educe the computational burden; how
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and y 3 as illustrated above. Again
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q In general, the number of paramet
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Similar to the fully structured mod
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ecause the former captures more of
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(Tsiamyrtzis and Karlis, 2004). The
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notation, the following recursive s
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5.2.2 The multivariate Poisson dist
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ypy ( ,0,0) = θ py ( − 1,0,0) y1
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ecurrence relationship ypy 1 ( 1 ,
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Raftery, 1998); identification of t
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maximization (EM) algorithm is appl
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the EM algorithms for use on very l
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Another alternative is to use the p
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5.3.5 Estimation for the multivaria
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j j E[ X 12 i | Yi , Z ij = 1, Φ ]
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5.4 Multivariate Poisson hidden Mar
Page 107 and 108: where P = Pr( S = k | S − 1 j), 1
Page 109 and 110: P jk n ∑ vˆ jk i= 2 = n m ∑∑
Page 111 and 112: E[ X | Y, u ( i) = 1, Φ ] = d = y
Page 113 and 114: The bootstrap method is a powerful
Page 115 and 116: eplications are generally sufficien
Page 117 and 118: Y and assumes a probability distrib
Page 119 and 120: For the case of two categorical var
Page 121 and 122: ejected. In this situation, the (sm
Page 123 and 124: the Poisson distribution is well su
Page 125 and 126: 0 1 2 3 4 Wild Buckwheat species109
Page 127 and 128: Table 6.4: The frequency of occurre
Page 129 and 130: 6.4.1 Results for the different mul
Page 131 and 132: Proportion 1.0 0.9 0.8 0.7 0.6 0.5
Page 133 and 134: -600 -650 -700 Loglikelihood -750 -
Page 135 and 136: Figure 6.7 illustrates the evolutio
Page 137 and 138: Table 6.8: Parameter estimates (boo
Page 139 and 140: common covariance and the four stat
Page 141 and 142: -600 -700 -800 Loglikelihood -900 -
Page 143 and 144: Table 6.11: Parameter estimates (bo
Page 145 and 146: 6.5 Comparison of the different mod
Page 147 and 148: loglikelihood providing at least a
Page 149 and 150: illustrates the contour plot of the
Page 151 and 152: (a) Independent Contour 1 Contour 2
Page 153 and 154: Karlis and Meligkotsidou (2006) dis
Page 155 and 156: The mean vector and the covariance
Page 157: The simple moments of B are polynom
Page 161 and 162: 7.5 Applications In addition to wee
Page 163 and 164: The estimated covariance matrix and
Page 165 and 166: The estimated covariance matrix (AI
Page 167 and 168: Table 7.6 Bacterial counts by 3 sam
Page 169 and 170: Table 7.8: Loglikelihood and AIC to
Page 171 and 172: (c) Finite mixture with the five co
Page 173 and 174: (i) Hidden Markov model with the fi
Page 175 and 176: CHAPTER 8 COMPUTATIONAL EFFICIENCY
Page 177 and 178: less time compared to the multivari
Page 179 and 180: 1400 1200 CPU time (1/100 second) 1
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Page 183 and 184: In this thesis, three species count
Page 185 and 186: In the applications of the HMMs, li
Page 187 and 188: indication of the relative goodness
Page 189 and 190: underlying data. The advantage of t
Page 191 and 192: 9.6 Further research We can present
Page 193 and 194: REFERENCES 1. Aas, K., Eikvil, L. a
Page 195 and 196: 17. Bicego, M., Murino, V. & Figuei
Page 197 and 198: 33. Descombes, X., Morris, R.D., Ze
Page 199 and 200: Department of Statistics, Universit
Page 201 and 202: 66. Li C.S., Lu J.C., Park J., Kim
Page 203 and 204: 85. Petrie T. (1969). Probabilistic
Page 205 and 206: 104. University of Manitoba, Depart
Page 207 and 208: z0
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threep2[i]
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loglike[nit]
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theta33
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} prob[g1+1, g2 + 1]
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theta131
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threep22[i]
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dens=matrix(0,nrow=T,ncol=N) alpha=
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# vˆ jk ( i) = P jk f ( y ; λ i
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